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1 Characterizing the Impact of Horizontal Heat Transfer on the Linear Relation of Heat Flow and Heat Production Ronald G. Resmini Department of Geography and GeoInformation Science College of Science George Mason University 4400 University Drive, MSN 6C3 Fairfax, VA 22030-4444 v: 703-470-3022 · e: rresmini@gmu.edu

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2 Horizontal heat transfer impacts the linear relation between heat flow and heat production Derived reduced heat flow and heat production depth values do not match actual values Reduced heat flows are higher; depth values are lower A method for characterizing the impact of horizontal heat transfer is presented here It is applicable to the uniform heat production model for two-dimensional heat transfer The method suggests an approach for correcting the effects of horizontal heat transfer Introduction

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3 Procedure (1 of 6) Construct a two-dimensional heat transfer model implementing the uniform heat production distribution The depth of the uniform heat production domains is constant across the two- dimensional heat transfer model space E.g., a model with five heat production domains is shown:

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4 Procedure (2 of 6) Obtain the steady-state temperature field for the model This is done analytically and numerically (see next slide) For every model, basal heat flow, Q f, is always 25.0 mW/m 2 Thermal conductivity, k T, is always 2.85 W/m.°C Calculate the linear heat flow/heat production (intercept and slope) parameters Note that they will not match the parameters of the forward model Repeat the previous steps...but decrease the depth to the base of the heat production domains...keeping the same heat production values and their spatial distribution I.e., construct another two-dimensional heat transfer model implementing the uniform heat production distribution but with a depth to the base of the heat production domains 1 km less than the previous model As before, the depth of the uniform heat production domains is still constant across the two-dimensional heat transfer model space

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5 A 1 W/m 3 A 2 W/m 3 A 3 W/m 3 A 4 W/m 3 A 5 W/m 3 0.00 W/m 3 125 km 35 km and x = 0x = L y = H y = 0 T = 0 The Problem Space Other models in this study are 250 kilometers in width (x-direction). y = b

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6 Procedure (3 of 6) This model with five heat production domains is shown overlain on the first model: 9 km 10 km

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7 Procedure (4 of 6) Repeat... decrease the depth to the base of the heat production domains... keeping the same heat production values and their spatial distribution I.e., construct yet another two-dimensional heat transfer model implementing the uniform heat production distribution but with a depth to the base of the heat production domains 1 km less than the previous model The depth of the uniform heat production domains is kept the same across the two-dimensional heat transfer model space Repeat until the heat production domain depths are zero. And as always, basal heat flow, Q f, is 25.0 mW/m 2 See next slide:

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8 Procedure (5 of 6)

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9 Procedure (6 of 6) Thirteen (13) such two-dimensional models were calculated and from which 13 pairs of heat flow/heat production linear parameters were derived: Depth to base of heat production region: 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.5, 0.1, 0 km A plot of intercept values (reduced heat flow), Y-axis, vs. values of true depth to the base of the heat production domains minus the slope (true depth – slope), X-axis, is constructed The plot has 13 points The plot has two linear portions A line is fit to the predominant linear portion (see next slides) The slope has the units of heat production, W/m 3 The intercept has the units of heat flow, mW/m 2

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10 Surface heat flow, Q, was calculated with the eq. below where T s is the surface temperature (T = 0 °C at y = 0), T d is the temperature at a depth of 1 km, k T is thermal conductivity, and d is a depth equal to 1 km: This method is used to maintain an internally consistent method of analyzing the results from both the analytical and numerical models and to simulate an actual field-measured thermal gradient and surface heat flux. Measuring Surface Heat Flow, Q

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11 Tools Pencil-’n-paper for analytical calculations Subsequently implemented in C and Pascal(!) www.freepascal.org The FlexPDE © finite element software system www.pdesolutions.com www.pdesolutions.com MS Excel ©

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12 Model 1a (Mod1a) 3.75 W/m 3 1.00 W/m 3 4.35 W/m 3 2.25 W/m 3 6.50 W/m 3 3.75 W/m 3 1.00 W/m 3 4.35 W/m 3 2.25 W/m 3 6.50 W/m 3 Distance, x direction, kilometers Heat Production, A, W/m 3 Results Model 1a (Mod1a) is comprised of five heat flow domains and is shown below:

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13 3.75 W/m 3 1.00 W/m 3 4.35 W/m 3 2.25 W/m 3 6.50 W/m 3 0.00 W/m 3 I.e., Model 1a is: 125 km 35 km

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14 Temperature at a depth of 1 km – Model 1a Analytical and numerical solutions are identical. Distance in the x-Direction (Km) Temperature (C) Analytical and Numerical Solutions

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15 Temperature at a depth of 35 km – Model 1a Analytical and numerical solutions are identical. Distance in the x-Direction (Km) Temperature (C) Analytical and Numerical Solutions

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16 Mod1a Heat Flow Vectors

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17 Model 1a (Mod 1a) The linear relation between surface heat flow (Q) and the heat production (A) of rocks exposed at the surface for Model 1a: True depth: 10 km; true basal heat flow, Q, 25.0 mW/m 2 Note that retrieved depth (slope) is <10 km (7.04 km) and the retrieved reduced heat flow (intercept) is >25.0 mW/m 2 (33.85 mW/m 2 ) Radiogenic Heat Production, A, W/m 3 Surface heat Flow, Q, mW/m 2

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18 Model 1a (Mod 1a) There are 12 more such plots for Model 1a... There are 13 pairs of Q-A depth and reduced heat flow values They are shown in the table below along with the calculation of (true depth, b – slope): Intercept values converge to 25.0 mW/m 2 Note X-axis Y-axis...on plot on next slide

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19 Model 1a (Mod 1a) The slope of the linear portion is 3.59 W/m 3 – remarkably close to 3.57 W/m 3 The intercept is also close to the true basal heat flux of 25.0 mW/m 2 Intercept, Reduced Heat Flow, Q, mW/m 2 True Depth – Slope (km) Ten (10) points form a sloping linear segment. The linear regression is based on the 10 points

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20 What is 3.57 W/m 3 ? HPE = Heat Producing Element It’s a distance-weighted average of the heat production values in the problem space: One only needs to know the HPE values along the surface of the problem space. (Readily obtainable from field and laboratory analyses.)

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21 This Intercept vs. Depth – Slope relationship is general; i.e., as will be shown next, it obtains when applied to several different models each with an average heat production value of 3.57 W/m 3. Models with other average heat production values are shown, too.

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22 More Analyses Eleven (11) sets of 13 two-dimensional model calculations were completed in the present study The problem domains are 125 km and 250 km in width All problem domains are 35 km in depth (y direction) The model configurations are given in the next several slides Many of the models were configured so that the average heat production value is 3.57 W/m 3 One model has unphysical heat production values to purposely yield an average of 0.0 W/m 3 The heat production domain widths are varied Q f, applied basal heat flow, is always 25.0 mW/m 2 The linear heat flow vs. (true depth – slope) relation is obtained

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23 HPE = Heat Producing Element

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24 Model 1 (Mod1) 3.75 W/m 3 1.00 W/m 3 4.35 W/m 3 2.25 W/m 3 6.50 W/m 3 3.75 W/m 3 1.00 W/m 3 4.35 W/m 3 2.25 W/m 3 6.50 W/m 3 Distance, x direction, kilometers Heat Production, A, W/m 3...similar to Mod1a except the problem space is longer in the horizontal direction

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25 Model 1 (Mod1) Radiogenic Heat Production, A, W/m 3 Surface heat Flow, Q, mW/m 2 The linear relation between surface heat flow (Q) and the heat production (A) of rocks exposed at the surface for Model 1: True depth: 10 km; true basal heat flow, Q f, 25.0 mW/m 2

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26 Model 1 (Mod1) Intercept, mW/m 2 Depth – Slope, km y = 3.57x + 23.215 R² = 1 20.0 22.0 24.0 26.0 28.0 30.0 0.000.250.500.751.001.251.501.752.00

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27 Model 2 (Mod2) Distance, x direction, kilometers Heat Production, A, W/m 3 3.75 W/m 3 1.00 W/m 3 4.35 W/m 3 2.25 W/m 3 6.50 W/m 3 3.75 W/m 3 1.00 W/m 3 4.35 W/m 3 2.25 W/m 3 6.50 W/m 3 3.75 W/m 3 1.00 W/m 3 4.35 W/m 3 2.25 W/m 3 6.50 W/m 3 3.75 W/m 3 1.00 W/m 3 4.35 W/m 3 2.25 W/m 3 6.50 W/m 3

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28 Model 2 (Mod2) Radiogenic Heat Production, A, W/m 3 Surface heat Flow, Q, mW/m 2 The linear relation between surface heat flow (Q) and the heat production (A) of rocks exposed at the surface for Model 2: True depth: 10 km; true basal heat flow, Q f, 25.0 mW/m 2

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29 Model 2 (Mod2) Depth – Slope, km Intercept, mW/m 2 y = 3.5789x + 23.211 R² = 1 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 0.000.501.001.502.002.503.003.50

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30 Model 3 (Mod3) Distance, x direction, kilometers Heat Production, A, W/m 3 3.50 W/m 3 2.75 W/m 3 6.50 W/m 3 1.25 W/m 3 4.65 W/m 3 3.50 W/m 3 2.75 W/m 3 6.50 W/m 3 1.25 W/m 3 4.65 W/m 3

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31 Depth – Slope, km Intercept, mW/m 2 y = 3.7344x + 23.134 R² = 1 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 40.0 0.000.501.001.502.002.503.003.504.00 Model 3 (Mod3)

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32 Model 4 (Mod4) Distance, x direction, kilometers Heat Production, A, W/m 3 3.50 W/m 3 2.75 W/m 3 0.00 W/m 3 -2.75 W/m 3 -3.50 W/m 3 3.50 W/m 3 2.75 W/m 3 0.00 W/m 3 -2.75 W/m 3 -3.50 W/m 3

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33 Model 4 (Mod4) Depth – Slope, km Intercept, mW/m 2 y =-0.1122x + 25.037 R² = 0.8657 20.0 22.5 25.0 27.5 30.0 0.000.200.400.600.801.001.201.401.60

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34 Model 5 (Mod5) Distance, x direction, kilometers Heat Production, A, W/m 3 6.50 W/m 3 1.00 W/m 3 3.25 W/m 3 2.75 W/m 3 6.50 W/m 3 1.00 W/m 3 3.25 W/m 3 2.75 W/m 3

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35 Model 5 (Mod5) Depth – Slope, km Intercept, mW/m 2 y = 2.5101x + 23.744 R² = 1 20.0 22.0 24.0 26.0 28.0 30.0 0.000.501.001.502.002.50

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36 Model 6 (Mod6) Distance, x direction, kilometers Heat Production, A, W/m 3 0.70 W/m 3 1.50 W/m 3 2.90 W/m 3 3.85 W/m 3 2.00 W/m 3 5.45 W/m 3 3.65 W/m 3 7.75 W/m 3 5.00 W/m 3 2.90 W/m 3 0.70 W/m 3 1.50 W/m 3 2.90 W/m 3 3.85 W/m 3 2.00 W/m 3 5.45 W/m 3 3.65 W/m 3 7.75 W/m 3 5.00 W/m 3 2.90 W/m 3

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37 Model 6 (Mod6) Radiogenic Heat Production, A, W/m 3 Surface heat Flow, Q, mW/m 2 The linear relation between surface heat flow (Q) and the heat production (A) of rocks exposed at the surface for Model 6: True depth: 10 km; true basal heat flow, Q f, 25.0 mW/m 2

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38 Model 6 (Mod6) Depth – Slope, km Intercept, mW/m 2 y = 3.5864x + 23.207 R² = 1 20.0 22.0 24.0 26.0 28.0 30.0 32.0 0.000.501.001.502.002.50

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39 Model 7 (Mod7) Distance, x direction, kilometers Heat Production, A, W/m 3 0.70 W/m 3 3.85 W/m 3 7.75 W/m 3 3.20 W/m 3 0.70 W/m 3 3.85 W/m 3 7.75 W/m 3 3.20 W/m 3

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40 Model 7 (Mod7) The linear relation between surface heat flow (Q) and the heat production (A) of rocks exposed at the surface for Model 7: True depth: 10 km; true basal heat flow, Q f, 25.0 mW/m 2 Radiogenic Heat Production, A, W/m 3 Surface heat Flow, Q, mW/m 2 y = 8.4601x + 28.797 R² = 0.9839 0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0 80.0 90.0 100.0 0.002.004.006.008.0010.00

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41 Model 7 (Mod7) Depth – Slope, km Intercept, mW/m 2 y = 3.6518x + 23.174 R² = 1 20.0 21.0 22.0 23.0 24.0 25.0 26.0 27.0 28.0 29.0 30.0 0.000.501.001.502.00

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42 Model 8 (Mod8) Distance, x direction, kilometers Heat Production, A, W/m 3 0.70 W/m 3 3.85 W/m 3 7.50 W/m 3 0.70 W/m 3 3.20 W/m 3 3.85 W/m 3 0.70 W/m 3 3.85 W/m 3 7.50 W/m 3 3.85 W/m 3 0.70 W/m 3 3.85 W/m 3 7.50 W/m 3 0.70 W/m 3 3.20 W/m 3 3.85 W/m 3 0.70 W/m 3 3.85 W/m 3 7.50 W/m 3 3.85 W/m 3

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43 Model 8 (Mod8) Radiogenic Heat Production, A, W/m 3 Surface heat Flow, Q, mW/m 2 The linear relation between surface heat flow (Q) and the heat production (A) of rocks exposed at the surface for Model 8: True depth: 10 km; true basal heat flow, Q f, 25.0 mW/m 2

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44 Model 8 (Mod8) y = 3.582x + 23.209 R² = 1 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 0.000.501.001.502.002.503.003.50 Depth – Slope, km Intercept, mW/m 2

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45 Model 9 (Mod9) Distance, x direction, kilometers Heat Production, A, W/m 3 2.15 W/m 3 6.25 W/m 3 1.60 W/m 3 2.15 W/m 3 6.25 W/m 3 0.50 W/m 3 2.15 W/m 3 6.25 W/m 3 2.15 W/m 3 6.25 W/m 3 2.15 W/m 3 6.25 W/m 3 1.60 W/m 3 2.15 W/m 3 6.25 W/m 3 0.50 W/m 3 2.15 W/m 3 6.25 W/m 3 2.15 W/m 3 6.25 W/m 3

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46 Model 9 (Mod9) Radiogenic Heat Production, A, W/m 3 Surface heat Flow, Q, mW/m 2 The linear relation between surface heat flow (Q) and the heat production (A) of rocks exposed at the surface for Model 9: True depth: 10 km; true basal heat flow, Q f, 25.0 mW/m 2

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47 y = 3.5805x + 23.21 R² = 1 20.0 22.0 24.0 26.0 28.0 30.0 32.0 34.0 36.0 38.0 0.000.501.001.502.002.503.003.504.00 Model 9 (Mod9) Depth – Slope, km Intercept, mW/m 2

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48 Model 10 (Mod10) Distance, x direction, kilometers Heat Production, A, W/m 3 2.15 W/m 3 6.25 W/m 3 1.60 W/m 3 0.50 W/m 3 2.15 W/m 3 6.25 W/m 3 1.60 W/m 3 0.50 W/m 3

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49 Model 10 (Mod10) Radiogenic Heat Production, A, W/m 3 Surface heat Flow, Q, mW/m 2 The linear relation between surface heat flow (Q) and the heat production (A) of rocks exposed at the surface for Model 10: True depth: 10 km; true basal heat flow, Q f, 25.0 mW/m 2

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50 Model 10 (Mod10) y = 3.5596x + 23.22 R² = 1 20.0 22.0 24.0 26.0 28.0 30.0 32.0 0.000.250.500.751.001.251.501.752.00 Depth – Slope, km Intercept, mW/m 2

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51 Seven Models Have A Mean A of 3.57 W/m 3 Models: 1, 1a, 2, 6, 8, 9, 10 From plots of Q vs. A for True Depth Equal to 10 km True basal heat flow, Q f, 25.0 mW/m 2

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52 Seven Models Have A Mean A of 3.57 W/m 3 Models: 1, 1a, 2, 6, 8, 9, 10 From plots of Intercept vs. Depth – Slope True basal heat flow, Q f, 25.0 mW/m 2

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53 The Four Additional Models Models: 3, 4, 5, 7 From plots of Q vs. A for True Depth Equal to 10 km True basal heat flow, Q f, 25.0 mW/m 2

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54 The Four Additional Models Models: 3, 4, 5, 7 From plots of Intercept vs. Depth – Slope True basal heat flow, Q f, 25.0 mW/m 2

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55 The Intercept vs. Depth – Slope Relationship Suggests a Method to Correct for or Model the Impact of Horizontal Heat Transfer on the Linear Relationship Between Surface Heat Flow and Heat Production.

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56 Intercept, Reduced Heat Flow, Q, mW/m 2 True Depth, b, – Slope (km) Q f =x 1 Q f =x 2 Q f =x m...

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57 Symbol Table

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58 A Comment on the Linear Regressions Linear regressions applied using MS Excel ® The Minitab statistical analysis software package xfitexy.c from Numerical Recipes in C, Press et al. The method of York and Evensen (2004) The last 3 methods attempted but not reported in the present study Regression method and results require additional investigation

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